Even a simple, stand-alone electronic garden timer can be controlled by your computer. Just pull it open, find the connections inside that come from the switch used to manually turn on the timer, and fit a socket to the case. Then you can just short the connection to trigger it to turn on.
In the photo you can see a cheapie Gardenmate timer tap that I got from Bunnings. At the bottom right is an RCA connector attached to the case, with a lead plugged in coming from the computer inside the house.